1st Grade - Gateway 2

The instructional materials for enVisionMATH California Common Core Grade 1 partially meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

The instructional materials present a Problem-Based Interactive Learning activity (PBIL) and a Visual Learning Bridge (VLB) within each lesson to develop conceptual understanding. However, the PBIL and VLB are teacher-directed and do not offer students the opportunity to practice conceptual understanding independently through the use of pictures, manipulatives, and models.

Overall, the instructional materials do not consistently provide students opportunities to independently demonstrate conceptual understanding throughout the grade level.

• In Topic 7 Lesson 7-3, the Overview of the PBIL states, “In this activity, children count groups of 10, record the numeral, and write how many 10s.” The Guided Practice directions state, “Count by 10s. Then write the numbers.” There is a picture number line as well as the three different ways to write that number above the Guided Practice problems for the students to reference. The Independent Practice directions state, “Count by 10s. Then write the numbers.” The picture number line continues above the Independent Practice problems for the students to reference. Students do not count independently, write how many 10s, or record the numeral of a given set of objects as they can copy the answers from the reference above.
• In Topic 9 Lesson 9-3, the Overview of the PBIL states, “In this activity, children will learn how to compare two-digit numbers using concrete materials.” The Independent Practice directions state, “Write the number of cubes. Circle is greater than or is less than.” Students do not compare two-digit numbers independently as the pictures are given for the numbers.
• In Topic 10 Lesson 101, the Overview of the PBIL states, “In this activity, children will use tens models to find sums of multiples of 10 up to 100.” The Independent Practice directions state, “Write the numbers to complete each number sentence.” Students do not use tens models independently to find sums of multiples of 10 as the pictures are given for the number sentences.

The instructional materials for enVisionMATH California Common Core Grade 1 partially meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied.

Each topic includes at least one Problem Solving lesson that can be found at the end of the topic. These lessons offer students opportunities to integrate and apply concepts and skills learned from earlier lessons. Within each individual lesson, there is a section titled, Problem Solving, where students practice the application of the mathematical concept of the lesson.

However, the instructional materials provide opportunities for working with the applications of mathematics through routine problems within the Problem Solving lessons and the Problem Solving section within all lessons.

The instructional materials have few opportunities for students to engage in non-routine application throughout the grade level. Examples of routine applications, where a solution path is readily available, are:

• In Topic 4 Lesson 4-10, students write a number sentence using addition or subtraction. Independent Practice problem 3 states, “Twan picks 9 berries. Then he picks 3 more. How many berries does Twan pick?”
• In Topic 6 Lesson 6-7, students write a number sentence using addition or subtraction. Independent Practice problem 4 states, “Dana won 12 spinning tops at the fair. She gave 6 of them to her friend. How many spinning tops does Dana have left?”
• In Topic 11 Lesson 11-5, students draw a picture to solve and write a subtraction sentence. Independent Practice problem 5 states, “A store had 70 toy cars. It sold 40 cars. How many cars does the store have left?” The problem has a blank subtraction number sentence below the story problem. The students fill in the blanks with the numbers.

The instructional materials reviewed for enVisionMATH California Common Core Grade 1 partially meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade level. Overall, the MPs are identified and used in connection to the content standards, but the materials do not always use the MPs to enrich the mathematics content. In the materials, the connections between the MPs and the content standards are not always clear.

• There are multiple places for finding the MPs in the materials: Content Guide in the Program Resources Tab, the beginning of each Topic, sometimes in the Math Background section within each Topic, and at the beginning of each lesson.
• Within each lesson there is a check list of MPs, but not all of the checked MPs are explicitly labeled within the lesson itself.
• In the Content Guide and the check lists, the MPs are labeled and addressed. Within enVisionMATH California Common Core Grade 1 lessons, the MP is abbreviated.
  • MP1 - Make Sense of Problems (no perseverance)
  • MP2 - Reason Quantitatively or Reason Abstractly (treated separately)
  • MP3 - Communicate or Critique the Reasoning of Others (treated separately)
  • MP4 - Model with Mathematics
  • MP5 - Use Appropriate Tools
  • MP6 - Attend to Precision
  • MP7 - Use Structure
  • MP8 - Check for Reasonableness or Make Generalizations (treated separately)
• In Math Background (page 1D) for Topic 1, the teachers note states, “Mathematical Practices Attend to Precision: To learn how to represent addition, children need many experiences moving among real-world addition stories, counters or pictures, and the symbols for addition.” This explanation is MP2 Reason Abstractly and Quantitatively, not MP6 Attend to Precision.
• In Lesson 10-3, there is a discussion and activity built around MP2 when adding 10, yet that MP is not identified. In the Visual Learning section, students answer, “How do you add 3 tens to 28 without seeing 28 cubes?” Followed by, “Why does the tens digit change? What digit stays the same? Explain.”
• In Lesson 12-3, MPs 2, 6, and 8 are identified. Within the lesson, there is a discussion and activity built around MP5."Should teacher steps be used to measure a pencil? What should the teacher use?" Yet that MP is not identified.

The instructional materials reviewed for enVisionMATH California Common Core Grade 1 partially meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

The Teacher Edition contains a Mathematical Practice Handbook which defines each math practice and includes question stems for each MP to help the teacher engage students. MP3 offers the following questions stems: “How can I use math to explain why my work is right?” “How can I use math to explain why other people’s work is right or wrong?” and “What questions can I ask to understand other people’s thinking?”

The materials label multiple questions throughout the material as MP3 or parts of MP3; however, those labeled have little information assisting teachers to engage students in constructing viable arguments or to critique the reasoning of others. The information that the materials provide is not specific and are often hints or reminders to give students while they are solving a problem.

Materials provide little assistance to teachers in engaging students in both constructing viable arguments and analyzing the arguments of others.

• In Lesson 1-5 there is little supporting commentary or questioning to assist teachers in helping students form or develop an explanation. For example, the prompt states, “As children say the sentences, have children ask their partner to check and see if they agree. If there is not agreement, have children discuss their opinions, talking about the workmat, the cubes, and the sentences to support their view.” This tag is in the Problem-Based Interactive Learning section. How students engage in deep mathematical conversation if they do agree is not explained further. When students disagree, only then critiquing others is encouraged.
• In Lesson 2-6 Extend, students explore the classroom looking for items to compare, “such as board erasers and erasable markers for the board or doors and windows. Encourage discussion.” Teachers are not given further direction as to whether students talk to each other or bring back their findings for a whole-group debrief.
• Topic 5 Communicate, page 163, states, "Encourage children to use examples as they discuss why some numbers are doubles and some are not.” There is no explicit mention of having children make sense of one another’s argument.
• In Lesson 6-2 Prevent Misconceptions, page 211, the teacher note says, "Children may think that the number they subtract to make 10 is the answer. Emphasize that they have to subtract a total of 6 counters, not 4, to find the answer to the problem.” This is a missed opportunity to get students in a discussion about subtraction strategies and critiquing others' reasoning when solving subtraction problems.
• In Lesson 11-5, pages 373-74, students are to solve the problem: “Jack took 50 berries from the basket. There are 40 berries left in the basket. How many berries were in the basket to begin with?” Teacher is directed that the “Drawing should show 8 boxes with 6 10-boxes underneath.” The teacher note for Constructing Arguments exercise 6, says: "Have children explain how they knew where to write the numbers in the subtraction sentence. Compare children’s drawings and have them explain how their drawing represents the problem.” This is a missed opportunity for students to share strategies used and for their classmates to critique their reasoning.
• In Topic 12 Math Background, the following explanation on how to use MP3 within the topic is provided: “Mathematical Practices: Communicate - Use language such as ‘This football is 2 straws long’ to emphasize that measurement is approximate.” The Topic 14 Math Background states, “Mathematical Practices: Communicate - To help children understand the meaning of graphs, emphasize making a class graph of favorite shoe colors shows which colors the class likes more or likes less. It is not about which color ‘wins’ or ‘loses.’” Critiquing others' arguments is not mentioned in either topic.